Questions — Pre-U Pre-U 9794/2 (176 questions)

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Pre-U Pre-U 9794/2 2020 Specimen Q10
4 marks Standard +0.8
10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.
  1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-4_624_897_1062_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates \(( 0.3,0.04 )\). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
    In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.
Pre-U Pre-U 9794/2 Specimen Q1
3 marks Moderate -0.3
1 Solve the equation $$x \sqrt { 32 } - \sqrt { 24 } = ( 3 \sqrt { 3 } - 5 ) ( \sqrt { 6 } + x \sqrt { 2 } )$$
Pre-U Pre-U 9794/2 Specimen Q2
4 marks Easy -1.2
2 You are given that \(\ln ( 12 ) = 2.484907\) and \(\ln ( 3 ) = 1.098612\), correct to 6 decimal places. Use the laws of logarithms to obtain the values of \(\ln ( 36 )\) and \(\ln ( 0.5 )\), correct to 4 decimal places. You must show your numerical working.
Pre-U Pre-U 9794/2 Specimen Q3
5 marks Standard +0.3
3 Show that $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \pi - x ) \cos 2 x \mathrm {~d} x = \frac { 1 } { 4 } + \frac { 3 } { 8 } \pi$$
Pre-U Pre-U 9794/2 Specimen Q4
6 marks Standard +0.3
4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).
Pre-U Pre-U 9794/2 Specimen Q5
9 marks Standard +0.3
5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).
  1. Find the radius of \(S\) and hence write down its cartesian equation.
  2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
    (b) Calculate the length of the minor arc joining these two points.
Pre-U Pre-U 9794/2 Specimen Q6
5 marks Standard +0.8
6
  1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
  2. Find an analogous result for three numbers in geometric progression.
  3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.
Pre-U Pre-U 9794/2 Specimen Q7
12 marks Standard +0.3
7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
  1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
  2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
  3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.
Pre-U Pre-U 9794/2 Specimen Q8
6 marks Standard +0.8
8
  1. Show that $$\tan x = \frac { 2 t } { 1 - t ^ { 2 } } \text { for } 0 \leq t < 1 , \text { where } t = \tan \frac { 1 } { 2 } x$$ and deduce that $$\sin x = \frac { 2 t } { 1 + t ^ { 2 } }$$
  2. Using the substitution \(t = \tan \frac { 1 } { 2 } x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 + \sin x } \mathrm {~d} x = \sqrt { 3 } - 1$$
Pre-U Pre-U 9794/2 Specimen Q9
11 marks Standard +0.3
9 A curve has equation $$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$
  1. Find the exact coordinates of the stationary points of \(y\).
  2. Determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
  3. Determine the nature of the stationary points on the curve.
Pre-U Pre-U 9794/2 Specimen Q10
12 marks Challenging +1.2
10
    1. By writing \(\sec x = \frac { 1 } { \cos x }\), prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x .$$
    2. Deduce that \(y = \sec x\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \sqrt { y ^ { 2 } - 1 } , \quad 0 \leq x < \frac { 1 } { 2 } \pi .$$
  2. A curve lies in the first quadrant of the cartesian plane with origin \(O\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-4_707_698_845_685} The normal to the curve at the point \(P ( x , y )\) meets the \(x\)-axis at the point \(Q\). The angle between \(O P\) and the \(x\)-axis is \(u\), and the angle between \(Q P\) and the \(x\)-axis is \(v\).
    1. If $$\tan v = \tan ^ { 2 } u$$ obtain a differential equation satisfied by the curve.
    2. The curve passes through the point \(( 2,1 )\). By solving the differential equation, find an equation for the curve in the implicit form $$\mathrm { F } ( x , y ) = C ,$$ where \(C\) is a constant that should be determined.
Pre-U Pre-U 9794/2 Specimen Q11
4 marks Challenging +1.2
11 Three light inextensible strings \(A C , C D\) and \(D B\), each of length 10 cm , are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-5_300_670_475_699} The ends \(A\) and \(B\) are fixed to points 20 cm apart on the same horizontal level. Two heavy particles, each of mass 2 kg , are attached at \(C\) and \(D\). The system remains in a vertical plane.
  1. Determine the tension in each string.
  2. The string \(C D\) is replaced by one of length \(L \mathrm {~cm}\), made of the same material. If the tension in \(A C\) is 50 N , show that \(L = 20 - 4 \sqrt { 21 }\).
Pre-U Pre-U 9794/2 Specimen Q12
5 marks Standard +0.3
12
  1. Whilst a helicopter is hovering, the floor of its cargo hold maintains an angle of \(30 ^ { \circ }\) to the horizontal. There is a box of mass 20 kg on the floor. If the box is just on the point of sliding, show by resolving forces that the coefficient of friction between the box and the floor is \(\frac { 1 } { \sqrt { 3 } }\).
  2. The helicopter ascends at a constant acceleration 0.5 g . If the cargo hold is now maintained at \(10 ^ { \circ }\) to the horizontal, determine the frictional force and the normal reaction between the box and the floor.
Pre-U Pre-U 9794/2 Specimen Q13
10 marks Standard +0.3
13 Professor Oldham wishes to illustrate and test Newton's experimental law of impacts. A ball is dropped from rest from a height \(h\) above a rigid horizontal board and rebounds to a height \(H\). The time taken to reach the height \(H\) after the first impact is \(T\). These quantities are recorded using very accurate measuring devices.
  1. Show that $$H = e ^ { 2 } h \quad \text { and } \quad T = e \sqrt { \frac { 2 h } { g } }$$ are predicted by Newton's law, where \(e\) is the coefficient of restitution between the ball and the board.
  2. If \(h = 180 \mathrm {~cm}\) and \(H = 45 \mathrm {~cm}\), determine \(T\) from these formulae. The experiment is repeated for initial heights \(h , 2 h , 3 h , \ldots , 15 h\) where \(h = 180 \mathrm {~cm}\). The corresponding rebound heights and times taken to reach that height after the first impact are recorded. The mean of the 15 rebound heights is found to be 3.3 m .
  3. Find the mean of the rebound heights predicted by Newton's law and give one reason why this differs from the experimental value. Professor Oldham is able to repeat the experiment on the surface of the moon using the same experimental set-up inside a laboratory.
  4. The mean of the rebound heights is unchanged, but the mean of the rebound times is substantially increased. Comment on these findings.
Pre-U Pre-U 9794/2 Specimen Q14
13 marks Standard +0.8
14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).
  1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
  2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$
Pre-U Pre-U 9794/2 2010 June Q1
3 marks Easy -1.8
Find the exact value of $$\int_1^4 \left(10x^2 - 3x^2\right) dx.$$ [3]
Pre-U Pre-U 9794/2 2010 June Q2
5 marks Standard +0.8
Solve the inequality $$\log_3(2x^2 - x) - \log_3(2x^2 - 3x + 1) > 1.$$ [5]
Pre-U Pre-U 9794/2 2010 June Q3
6 marks Standard +0.3
An arithmetic progression has 13th term equal to 60 and 31st term equal to 141.
  1. Find the first term and common difference of the progression. [3]
A second arithmetic progression has first term 1.5 and common difference 3.
    1. Write down the first four terms of each progression. [1]
    2. Prove that the two progressions have an infinite number of terms in common. [2]
Pre-U Pre-U 9794/2 2010 June Q4
6 marks Standard +0.3
  1. Show that $$\cos^4 x - \sin^4 x = 2\cos^2 x - 1.$$ [2]
  2. Hence find the solutions of $$\cos^4 x - \sin^4 x = \cos x,$$ where \(0° \leqslant x \leqslant 360°\). [4]
Pre-U Pre-U 9794/2 2010 June Q5
9 marks Standard +0.8
It is given that $$y = \frac{1}{x+1} + \frac{1}{x-1},$$ where \(x\) and \(y\) are real and positive, and \(i^2 = -1\).
  1. Show that $$x = \frac{1 \pm \sqrt{1-y^2}}{y} \quad \text{and} \quad y \leqslant 1.$$ [4]
  2. Deduce that $$xy < 2.$$ [2]
  3. Indicate the region in the \(x\)-\(y\) plane defined by $$y \leqslant 1 \quad \text{and} \quad xy < 2.$$ [3]
Pre-U Pre-U 9794/2 2010 June Q6
10 marks Standard +0.3
  1. Express \(\frac{x-1}{x^2+2x+1}\) in the form \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\), where \(A\) and \(B\) are integers. [2]
  2. Find the quotient and remainder when \(2y^2 + 1\) is divided by \(y + 1\). [2]
  3. A curve in the \(x\)-\(y\) plane passes through the point \((0, 2)\) and satisfies the differential equation $$(2y^2 + 1)(x^2 + 2x + 1)\frac{dy}{dx} = (x - 1)(y + 1).$$ By solving the differential equation find the equation of the curve in implicit form. [6]
Pre-U Pre-U 9794/2 2010 June Q7
12 marks Standard +0.3
Let \(y = (x - 1)\left(\frac{2}{x^2} + t\right)\) define \(y\) as a function of \(x\) (\(x > 0\)), for each value of the real parameter \(t\).
  1. When \(t = 0\),
    1. determine the set of values of \(x\) for which \(y\) is positive and an increasing function, [3]
    2. locate the stationary point of \(y\), and determine its nature. [2]
  2. It is given that \(t = 2\) and \(y = -2\).
    1. Show that \(x\) satisfies \(f(x) = 0\), where \(f(x) = x^3 + x - 1\). [1]
    2. Prove that \(f\) has no stationary points. [2]
    3. Use the Newton-Raphson method, with \(x_0 = 1\), to find \(x\) correct to 4 significant figures. [4]
Pre-U Pre-U 9794/2 2010 June Q8
14 marks Standard +0.8
The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).
  1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
  2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
    1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
    2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]
Pre-U Pre-U 9794/2 2010 June Q9
15 marks Challenging +1.2
  1. Show that $$\int x^n \ln x \, dx = \frac{x^{n+1}}{(n+1)^2}\left((n+1)\ln a - 1\right) + \frac{1}{(n+1)^2},$$ where \(n \neq -1\) and \(a > 1\). [6]
    1. Determine the \(x\)-coordinate of the point of intersection of the curves \(y = x^3 \ln x\) and \(y = x \ln 2^x\), where \(x > 0\). [2]
    2. Find the exact value of the area of the region enclosed between these two curves, the line \(x = 1\) and their point of intersection. Express your answer in the form \(b + c \ln 2\), where \(b\) and \(c\) are rational. [4]
  3. The curve \(y = (x^3 \ln x)^{0.5}\), for \(1 < x < e\), is rotated through \(2\pi\) radians about the \(x\)-axis. Determine the value of the resulting volume of revolution, giving your answer correct to 4 significant figures. [3]
Pre-U Pre-U 9794/2 2010 June Q10
9 marks Standard +0.8
A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).
  1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
  2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]